Theorem sucexeloni 7799

Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7801 does not require ax-un 7727. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)

Assertion

Ref	Expression
sucexeloni	⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloni

Step	Hyp	Ref	Expression
1		eloni 6373	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsuci 7798	. . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
4		elex 3491	. 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V)
5		elong 6371	. . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
6	5	biimparc 478	. 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On)
7	3, 4, 6	syl2an 594	1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2104  Vcvv 3472  Ord word 6362  Oncon0 6363  suc csuc 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366  df-on 6367  df-suc 6369

This theorem is referenced by:  onsuc  7801  1on  8480  2on  8482